Greatest Common Divisor (GCD) of 75 and 195
The greatest common divisor (GCD) of 75 and 195 is 15.
What is the Greatest Common Divisor (GCD)?
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is useful in simplifying fractions, finding common factors, and in number theory.
How to Calculate the GCD of 75 and 195?
We use the Euclidean algorithm, which involves the following steps:
- Divide the larger number by the smaller number.
- Replace the larger number with the smaller number and the smaller number with the remainder from the division.
- Repeat this process until the remainder is zero.
- The non-zero remainder just before zero is the GCD.
Step-by-Step Euclidean Algorithm
| Step | Calculation |
|---|---|
| 1 | 75 ÷ 195 = 0 remainder 75 |
| 2 | 195 ÷ 75 = 2 remainder 45 |
| 3 | 75 ÷ 45 = 1 remainder 30 |
| 4 | 45 ÷ 30 = 1 remainder 15 |
| 5 | 30 ÷ 15 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 173 and 42 | 1 |
| 123 and 78 | 3 |
| 67 and 142 | 1 |
| 107 and 184 | 1 |
| 119 and 90 | 1 |